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September 21, 2008

Back and Catching Up

Posted by Urs Schreiber

After my stay at ESI in Vienna # I spent a week hiking in the Alps with my family. Amazing how much one can miss in one week. Below a list of things I need to catch up with.

The next two weeks I’ll be in Zagreb, visiting Zoran Škoda and Igor Baković with whom we are working on a project on nonabelian differential cohomology (with David Roberts, Hisham Sati and Danny Stevenson). I am hoping to be able to say more about this in a while. Am planning to talk about something at least related at Higher Structures 2008 in November.

Here the list of things to read which I missed last week. And even this list is gonna be incomplete, as I am running out of time…

And as Jim Stasheff kindly points out, I should blog about the highly interesting new piece by Sergei Gukov and Edward Witten: Branes and Quantization

Abstract:
The problem of quantizing a symplectic manifold (M,\omega) can be formulated in terms of the A-model of a complexification of M. This leads to an interesting new perspective on quantization. From this point of view, the Hilbert space obtained by quantization of (M,ω)(M,\omega) is the space of (Bcc,B′)(B_{cc},B') strings, where BccB_{cc} and B′B' are two A-branes; B′B' is an ordinary Lagrangian A-brane, and Bcc is a space-filling coisotropic A-brane. B′B' is supported on MM, and the choice of \omega is encoded in the choice of BccB_{cc}. As an example, we describe from this point of view the representations of the group SL(2,ℝ)SL(2,\mathbb{R}). Another application is to Chern-Simons gauge theory.

More later.

Posted at September 21, 2008 2:21 PM UTC

TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1797

Finally, I have just been alerted to apply for Autumn 2009 at the MPI Bonn where Stephan Stolz and Peter Teichner will organize a meeting on functorial field theory. Application requires a PhD, but there are also graduate programs.

Various aspects of homotopy algebras such as A-infinity, L- infinity, and open-closed homotopy algebras are currently the object of research by a large number of mathematicians. These algebras are closely linked to a variety of topics in Mathematical physics, for example, open string theory, closed string theory, gauge theory. This session will provide a forum for reports on current developments in both areas - algebraic structures as well as physical implications.

We particularly encourage graduate students and post-docs to report on their current research on these topics.

We hope to have at least one lecture that will present the overall picture that relates these subjects. If you are interested in giving such a survey, please let us know.

Respond to both of us, please, if you are even tentatively interested or know for sure you will not be able to participate. Either way, let us know of
other possible participants.

Re: Back and Catching Up

It looks like taxpayers in the USA will now pay at least $700 billion to help bankers who couldn’t balance their books.

I hope all you readers from other countries express your thanks by taking me out for a coffee the next time we meet!

On the bright side, California has finally decided on a budget, almost 80 days late thanks to wrangling between Democrats and Republicans — but in time for my wife and me to avoid having our pay cut to minimum wage. So, maybe you should buy someone poorer that coffee.

Sorry to intrude with real-world news, but this was quite a week. I wouldn’t want people looking back at this blog someday to think we were completely oblivious.

Re: Back and Catching Up

An open question is where economy-mindedness comes from. I wonder if the associated motivations and ways of thinking are only some kind of stabilized deviation, perhaps developed in an analogous way like “tricksters”?

Re: Back and Catching Up

New Scientist: “Banks pay enormous sums to lure researchers away from other areas of science and set them to work building complex statistical models that supposedly tell the bankers about the risks they are running. So why didn’t they see what was coming?”

Re: Back and Catching Up

Hi Urs,

Not that i want to steer you guys away from the more “practical matters”… but, a discussion about the last two papers you cited (on your post, namely Hollands’ and Witten’s) would really interest me — in fact, i’ve been waiting for some “resonance” on the blogosphere for some time (i was moving and dealing with all sorts of “changing universities”-issues and couldn’t bring myself to do it).

Hollands proposes to axiomatize Euclidean quantum field theory in terms of the operator product expansion (OPE) of its poinlike fields. In fact, he goes one step further and suggests to take the OPE to be more basic than the concept of quantum field itself. So essentially his suggestion boils own to studying certain algebras which depend on parameters (spacetime points) and satisfy a refined associativity condition depending on these parameters (the OPE factorization property).

Of course essentially this is done one way or another commonly in chiral 2-dimensional and especially in chiral 2-dimensional conformal QFT, where this is essentially the picture of vertex operator algebras.

As we have recently mentioned a couple of times (first here), vertex operator algebras are now understood as precisely the algebras for the operad whose morphisms are conformal spheres with nn conformally parameterized incoming and one conformally parameterized outgoing insertion point which depend holomorphically on these insertion points. So VOAs have now a good interpretation in terms of Euclidean functorial QFT (“FQFT”).

In particular, the factorization property in this context is just the obvious functoriality, i.e the operad-algebra property, i.e. the respect for the compositition of spheres along their parameterized insertion points.

For that reason it seems to me that Hollands’ approach might benefit from a investigation into its relation to representations of the corresponding higher dimensional cobordism operads/categories.

Also, Hollands announces upcoming work with Wald on the generalization of their approach to the non-Euclidean Lorentzian setup. That should be more interesting, since it should involve in particular passing from chiral QFT to full QFT. From the hints he gives in the present article I came to wonder what he might have on top of the standard AQFT picture of nets of local operator algebras, which gives the OPE picture (in the Lorentzian case) by passing to “stalks” i.e. to field algebras living over regions asymptoting to points.

Actually recently, I sat in a talk Wald gave on this work with Hollands. But I didn’t pay much attention, because after the introduction of this talk I had the insight which has now become AQFT from FQFT# and I was busy the rest of the time drawing pictures on scratch paper… ;-)

Re: Back and Catching Up

Although i did like your paper (AQFT <=> FQFT) and even have a little picture along its lines in my thesis (which i still need to “fix” in order to post), it’s too bad you missed Wald’s talk: i’m pretty curious now! :-)

In fact, since Wald’s previous paper on this subject, where he already mentioned using the OPE as the field algebra, i have been waiting for some follow up… Hollands is the first installment, albeit Euclidian.

In particular, i’m curious to see what will happen to non-perturbative issues, e.g., symmetry breaking. In this “Euclidian installment”, Hollands points out that the perturbative series (in the coupling constant) can be “bootstrapped” and is convergent (thus, being a Taylor series). I don’t think this will be true in the Lorentzian case, where the series should be asymptotic (rather than convergent), otherwise the theory would have only one phase. I’m interested to see what’s going to happen to the coupling constants in different phases, i.e., whether they can be related to each other or not — mainly so because of this Hochschild cohomology that guides the terms of the perturbative expansion.